gamma distribution

#1
Show that the sum of n independent identically distributed exponential random variables with parameter theta has a gamma distribution and give its parameters.

I am not even sure what this is talking about. Can anyone give me a starting point?
 

BioStatMatt

TS Contributor
#2
Try using the MGF method

OK, This kind of question indicates that might be taking a mathematical statistics class. If you are not, this answer might not work for you. I will assume this because I'm not sure how to answer at a lower level than that course.

You know that the gamma and exponential distributions are frequency distributions like the normal distribution. What you are asked to do is find the distribution of a sum of random variables. That is:

Y = X1 + X2 + ... + Xn

where X1, X2, ..., Xn are independent and are distributed identically, i.e. all the x's follow the exponential distribution. Your task is to show that Y has a gamma distribution and list its parameters. This is called a transformation.

There are several ways to transform random variables, the Jacobian method, the CDF method, but the most straight forward technique to use here would be the moment generating function(MGF) method.

Recall that the MGF of a sum of independent random variables is the product of the individual MGFs. That is:

MGFy = MGFx1 * MGFx2 * ... * MGFxn

Since the MGF uniquely identifies the distribution of a random variable, you should see that, in this case, the MGF of Y will be the MGF of a gamma distribution.

Hint: Look up what the MGF of the gamma distribution should look like and what its parameters are.

I hope this helps. I will monitor this thread for a while, if you have any trouble.

~Matt